Solve the exponential equation for $x$. 5 4 x + 3 25 9 − x = 5 2 x + 5 \dfrac{5\^{4x+3}}{25\^{ 9-x}}=5\^{ 2x+5} $x=$
Explanation: The strategy Let's write $25$ in base $5$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $5$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 5 4 x + 3 25 9 − x = 5 4 x + 3 ( 5 2 ) 9 − x = 5 4 x + 3 5 18 − 2 x = 5 4 x + 3 − ( 18 − 2 x ) = 5 6 x − 15 ( 25 = 5 2 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned} \dfrac{5\^{4x+3}}{25\^{ 9-x}}&=\dfrac{5\^{ 4x+3}}{(5^2)\^{ 9-x}}&&&&(25=5^2) \\\\\\\\ &=\dfrac{5\^{ C{4x+3}}}{5\^{ {18-2x}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=5\^{ C{4x+3} \ - \ ({18-2x})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=5\^{ 6x-15} \end{aligned} Solving the equation We obtain the following equation. 5 6 x − 15 = 5 2 x + 5 5\^{ 6x-15}=5\^{ 2x+5} Now we can equate the exponents and solve for $x$. $\begin{aligned} 6x-15 &=2x+5\\\\ x &= 5\end{aligned}$ The answer The answer is $x=5$. You can check this answer by substituting $\it{x=5}$ in the original equation and evaluating both sides.